318 XV NORMED ALGEBRAS AND SPECTRAL THEORY (b) If x, y e A are such that \\c**y e-t*\\ ^ k\\y\\ (k a constant >0) for all £ 6 C, show that xy = yx. (Use Liouville's theorem and the Taylor expansion of the function £ t-te^ye"** at £ = 0.) (c) If \\x2\\ = ||je||2 for all x e A, show that A is commutative. (Observe that we have p(x)= \\x\\ for all xeA, and apply this relation to e^ye'1**; then use (a) and (b).) (d) Show that, if there exists a constant k > 0 such that \\yx\\ <J k\\xy\\ for all x,y in A, then A is commutative. (Apply the inequality with e~**x in place of x and e;*y in place of y, and use (b).) (e) Let a e A be such that for all £ e C and all x e A. Prove that a belongs to the center of A. (Prove that, for each f E C and all integers n^n0 (where n0 depends on £)» for all y e A; then let n tends to •+• oo and use (b).) 15. Let A be a Banach algebra with unit element e, and let C be a commutative Banach subalgebra containing e. Show that there exists a commutative Banach subalgebra B of A containing C such that SpB(x) = SPA (x) for each x e C. 3. CHARACTERS AND SPECTRUM OF A COMMUTATIVE BANACH ALGEBRA. THE GELFAND TRANSFORMATION Let A be a commutative algebra (over C). A character of A is defined to be any homomorphisrn x °f the algebra A into C which is not identi- cally zero. Because #(1*) = %x(x) for all scalars A, this condition is equivalent to #(A) = C. If A has a unit element e ^ 0, then %(e) ^ 0 (otherwise x(x) =x(ex) = x(e)x(x) — 0 for all x e A); it follows that x(e) = 1> because %(e)2 =^(e2) = x(e)< An ideal m ^ A in A is said to be maximal if there exists no ideal n such that m T»£ n, n ^ A and m c n. If A has a unit element, m is maximal if and only if A/m is a (commutative) j?eW. (15.3.1) Let A be a commutative Banach algebra with unit element e =£ 0. (i) For each x e A and each character x of A, we have x(x) G SpA(:x;). (ii) Every character x of A is a continuous linear form with norm 1. (iii) The mapping XI~>X~1(^) ^ a bijection of the set of characters of A onto the set of maximal ideals of A (which are therefore closed). Assertion (i) is a particular case of (15.2.8(i)). It follows (15.2.4(iii)) that \x(x)\ ^ \\x\\, which shows (5.5.1) that ^ is a continuous linear form with norm gl; and since %(e) = 1 and \\e\\ = 1, we have \\x\\ = 1. Finally, since(c).)